Statistics – Spring 2008
Lab #6 – Mediation and SEM
Defined:
Variables:
Relationship:
Example:
Assumptions:
Testing interrelationships amongst variables
Variables are continuous or categorical
Structure amongst variables
What is relationship between provocation, anger, aggression, etc.
If linear, then linear assumptions. If categorical, then multicollinearity.
1. Mediation
 SPSS can calculate the Baron/Kenny approach using three separate linear regressions, but SPSS can not
calculate the Sobel Test.
 A macro on the following website allows SPSS to calculate Baron/Kenny and Sobel simultaneously.
 Here are the instructions for using the macro:
1. Go to this website – http://www.comm.ohio-state.edu/ahayes/sobel.htm (also listed on PsychWiki)
2. Download the macro to your computer by clicking on “Click here to download SPSS syntax file”.
3. Open the macro by double-clicking on it.
4. The new “syntax” window that opens up is the macro.
5. The blank data editor that opens up is where you need to put the data you want to analyze.
6. For our example, I am going to analyze three variables: commit1, liborcon, rel_person. Thus, I open our
“legal beliefs” data file, copy the variables, and then paste the variables into the blank data editor.
7. You “activate the macro” by highlighting everything in the “syntax” window, and clicking the blue triangle
underneath the “Graphs” pull-down menu. The information displayed in the “Output” window will be a
duplicate of the syntax code for the macro.
8. You are now ready to start conducting mediation.
9. In the syntax window, copy/paste the following text: SOBEL y=yvar/x=xvar/m=mvar/boot=z.
10. In the place of “yvar”, put the SPSS variable name of your DV
11. In the place of “xvar”, put the SPSS variable name of your IV
12. In the place of “mvar”, put the SPSS variable name of your mediator.
13. In the place of “z”, put 1000. This tells the macro to run bootstrapping. I will explain this later.
14. As a concrete example, I want to test the relationship between: commit1, liborcon, rel_person.
15. More specifically, I want to test the relationship between commit1 and liborcon, and then see if rel_person
is a mediator. To put that into “hypothesis” wording, I am starting with the finding that there is a positive and
significant relationship between being “liberal versus conservative” and belief about “What percent of people
brought to trial did in fact commit the crime?”. In other words, being more conservative leads to a higher
percentage of defendants that people believe are guilty by simply being brought to trial. The hypothesis is
whether or not being a religious person (rel_person) mediates that relationship. In other words, maybe being
“liberal or conservative” leads to higher percentage for “commit1” BECAUSE of religiousocity.
16. To put that into a graphical form:
Religious
Liberal-Con
commit1
1
17. To put that relationship into “syntax” wording, I type the following into the syntax window:
y=commit1/x=liborcon/m=rel_person/boot=1000.
18. I highlight only that new line, and click the blue arrow.
19. The output window will show the analysis:

Pasted below is the entire output:
a. DIRECT AND TOTAL EFFECTS are the steps for the Baron and Kenny approach.
In the first step of analysis, there was a significant relationship between IV & DV (p = .0860).
In the second step of analysis, there was a significant relationship between IV & M (p = .0000).
In the final step of the analysis, there was a significant relationship between M & IV (p = .0293),
while the relationship between IV & DV became non-significant
(p = .3411).
b. INDIRECT EFFECT is the sobel test. The significant value (p = .0424) tells you that “rel_person” is a
significant mediator between the IV and DV
c. BOOTSTRAPPING tells you the same thing as the sobel test, but: (1) it calculates the analysis by first
“bootstrapping”, which is a type of robust technqiue to control for non-normality, and (2) it gives you
output in the form of “confidence intervals” instead of “p values”. I suggest reporting the Sobel test,
and ignore the others tests (Baron/Kenny, bootstrapping).

Here is how you report the sobel test:
a. There was a significant initial relationship between the independent variable and dependent variable (
= .10, p = .086) that was non-significant after controlling for the mediator ( = .06, p = .341) which
indicates level of religiousness mediates the relationship between the independent variable and
dependent variable.
b. FYI – I got the effect size () for the relationship between the “IV and DV” by conducting bivariate
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

regression (which is the same as bivariate correlation), and I got the effect size () for the relationship
between the “IV and DV while controlling for the mediator” by including those three variables in a
linear regression analysis and looking at the standardized beta for “liborcon” to “commit1” (in other
words, the linear regression in which you put both “liborcon” and “rel_person” as IV into the analysis
will report the unique effect of each variable on the DV, so the unique effect of “liborcon” on the DV
is what you are looking for to report as the non-significant effect after controlling for the mediator).
After you conduct your “confirmatory” analysis of your hypothesized relationship amongst the variables, you
then do exploratory analysis, such as first testing alternative models (that you want to NOT WORK) because
then it strengthens your hypothesis by having your hypothesis be the only relationship amongst the variables
that does work. (By “work”, we mean finding a significant mediational relationship). With three variables,
there are six total permutations. You can run all six if you want to. Some people only run the permutation in
which you switch the M and DV (called “reverse” mediation). If none of the alternative models work, you
report that in the following way: “No alternative models showed a significant mediational relationship
amongst the variables.”
Also, notice that you typically want the bivariate relationships amongst your variables to be significant. Thus,
I usually look at the bivariate correlations before conducting mediational analysis.
2. Structural Equation Modeling
 I am going to show you the basic steps involved in SEM so that you are exposed to this type of statistical
technique, and so that you understand the basic functions of the analysis.
 SEM is the ultimate type of statistical analysis because you can include any type of variable (e.g., continuous,
categorical, dichotomous) and look at any type of relationship (e.g., any permutation).
 SEM is the most complicated type of statistical analysis. Entire courses are devoted to understanding SEM, so
I can’t cover everything there is to know about SEM; instead I will provide the essential information.
 If you want to conduct SEM in the future, see me after the class is over and we can talk about how to get the
software (EQS) and I can show you how to conduct the analysis in-depth.
 For simplicity sake, I am going to show you the data from one of my studies on intergroup aggression. We
were interested in cycles of retributive violence, and how conflict spreads beyond the initial actors who started
the dispute.
 We asked participants to tell us a time when a fellow ingroup member was harmed by an outgroup member.
(the “ingroup” and “outgroup” could be any type of group, such as religion, ethnicity, family, gang, etc).
 We then asked participants to rate the experience along certain dimensions, such as
a. their bond with the ingroup victim (identification),
b. the perceived bond between outgroup members (entitativity),
c. their emotional reactions (anger) and
d. their feelings of retribution (toward the perpetrator, and towards group members of the perpatrator.
 Our hypothesis was that
a. “Identification” was a pre-requisite to seeking retribution, and that it would influence retribution via
emotional reactions like anger
b. “Entitativity” was a pre-requisite to seeking retribution, but since it is a cognitive variable (and not an
emotional variable), it would not influence retribution via emotional reactions like anger.
c. Identification is a “motivator” so it will influence retribution toward everyone -- both the perpetrator
and anyone involved (e.g., group member of the perpetrator)
d. Entitativity is a cognitive perception of the outgroup, so it will ONLY influence retribution toward the
group (but not the perpetrator)
e. In sum, the reason why cycles of retributive violence continue is because of your attachment to the
victim (identification) and perception of outgroup bonds (entitativity) that produces a desire to seek
retribution toward the entire outgroup.
f. The underlying assumption was that if you seek retribution toward the outgroup, other outgroup
members who feel attached to the new victim (identification) will then seek retribution toward your
group (entitativity). Thus, both sides undergoe the same process, and the cycle of violence continues as
each side seeks more retribution back and forth and back and forth and etc.
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

The way you conduct SEM to test this hypothesized relationship involves three steps.
First, draw a path model of the hypothesized relationships:
a. Below is a graphical representation I drew in “Microsoft Word”. I drew this graphical representation
simply to help you understand the path model. This step is not necessary to conduct SEM. The next
picture below show you how to draw the path model in EQS. That step is necessary to conduct SEM.
b. Here is the path model in EQS. The EQS program uses SPSS files, so the following variable names are
from my SPSS file. The arrows represent the direction of the relationship. The “e” is the error term.
Notice that there is not a bidirectional arrow between the two DVs. This is because you insert that
relationship manually in the next step of the process.
IN_ID3
INTENT_4
ANGRY_C
OUT_E3

E1
E5
INTENT_5
E2
Second, EQS converts the path model to formulas and syntax. Notice that the “Equations” look similar to
linear regression equations in which the DV is predicted from certain IVs and error terms. Notice also that
under “Covariances”, I manually inputted the bidirectional relationship between the DVs (e.g., E1,E2 = *;)
/TITLE
Model built by EQS 6 for Windows
/SPECIFICATIONS
DATA='c:\documents and settings\doug stenstrom\desktop\new folder\100cases.ess';
VARIABLES=7; CASES=100;
METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW;
/LABELS
V1=INTENT_4; V2=INTENT_5; V3=IN_ID3; V4=OUT_E3; V5=ANGRY_C;
V6=SAD_C; V7=EMOTIN15;
/EQUATIONS
V1 =
*V4 + *V5 + E1;
V2 =
*V4 + *V5 + E2;
V5 =
*V3 + E5;
/VARIANCES
V3 = *;
V4 = *;
E1 = *;
E2 = *;
E5 = *;
/COVARIANCES
V3,V4 = *;
E1,E2 = *;
/PRINT
4
EIS;
FIT=ALL;
EFFECT=YES;
TABLE=EQUATION;
/LMTEST
PROCESS=SIMULTANEOUS;
SET=PVV,PFV,PFF,PDD,GVV,GVF,GFV,GFF,
BVF,BFF;
/WTEST
PVAL=0.05;
PRIORITY=ZERO;
/END

Third, you run the analysis and get output. The output below is only a small percentage of the total output.
The entire output is 10 pages. Each piece of the output is informative, but I just want to show you the essential
information:

This part of the output tells you whether your data are “normal”. If the “Normalized estimate” is above 3, then
your data are non-normal. In this case, our data are “normal”.
MULTIVARIATE KURTOSIS
--------------------MARDIA'S COEFFICIENT (G2,P) =
NORMALIZED ESTIMATE =

4.1230
2.4516
This part of the output tells you how the model fits your data according to various criteria. Remember that
evaluating multiple fit indices simultaneously is recommended because different indices assess different
aspects of goodness-of-fit, and there is not always agreement on what constitutes good fit, so satisfactory
models should show consistently good-fitting results on many different indices. Here are the benchmarks for a
“good” fitting model: Chi-square of p < .05; Ratio of x2/df < 3; SRMR < .08; CFI > .95.
See the output below for the fit of our model: x2=1.03, p =.794, x2/df =.34, SRMR =.03, CFI =1.00.
GOODNESS OF FIT SUMMARY FOR METHOD = ML
INDEPENDENCE MODEL CHI-SQUARE
INDEPENDENCE AIC =
MODEL AIC =
72.65479
-4.97080
=
92.655 ON
INDEPENDENCE CAIC =
MODEL CAIC =
10 DEGREES OF FREEDOM
36.70359
-15.75616
CHI-SQUARE =
1.029 BASED ON
3 DEGREES OF FREEDOM
PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS
.79419
THE NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION IS
FIT INDICES
----------BENTLER-BONETT
NORMED FIT INDEX =
.989
BENTLER-BONETT NON-NORMED FIT INDEX =
1.079
COMPARATIVE FIT INDEX (CFI)
=
1.000
BOLLEN
(IFI) FIT INDEX
=
1.022
MCDONALD (MFI) FIT INDEX
=
1.010
LISREL
GFI FIT INDEX
=
.996
LISREL
AGFI FIT INDEX
=
.979
ROOT MEAN-SQUARE RESIDUAL (RMR)
=
.133
STANDARDIZED RMR
=
.028
ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA)
90% CONFIDENCE INTERVAL OF RMSEA (
.000,
=
1.030.
.000
.108)
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
This part of the output tells you the effect sizes for the relationships amongst your variables.
STANDARDIZED SOLUTION:
INTENT_4=V1
INTENT_5=V2
ANGRY_C =V5

=
=
=
.452*V5
.314*V5
.247*V3
R-SQUARED
+ .159*V4
+ .199*V4
+ .969 E5
+ .873 E1
+ .924 E2
.237
.145
.061
This part of the output tells you if there are any relationships (e.g., lines between variables) that should be
included or excluded.
MULTIVARIATE WALD TEST BY SIMULTANEOUS PROCESS
CUMULATIVE MULTIVARIATE STATISTICS
---------------------------------STEP PARAMETER
CHI-SQUARE D.F. PROBABILITY
---- ----------- ---------- ---- ----------1
V4,V3
3.548
1
.060
UNIVARIATE INCREMENT
-------------------CHI-SQUARE PROBABILITY
---------- ----------3.548
.060
LAGRANGE MULTIPLIER TEST (FOR ADDING PARAMETERS)
ORDERED UNIVARIATE TEST STATISTICS:
HANCOCK
CHI3 DF
PARAMETER
NO
CODE
PARAMETER
SQUARE
PROB.
PROB.
CHANGE
------------------------ -------- --------1
2 11
V5,V4
.458
.499
.928
.045
2
2 11
V2,V3
.337
.562
.953
.064

STANDARDIZED
CHANGE
-------.014
.013
After collecting all available information from the output, you write-up the results using the following path
diagram and text:
.25**
Identification
Anger
composite
.23*
Entitativity
.45***
.31***
.20*
Retribution
towards the
Perpetrator
.56***
Retribution
towards the
Group
Structural equation model of Study 1 variables including ingroup identification, outgroup entitativity,
anger, retribution toward initial perpetrator, and retribution toward the entire outgroup. Paths with
single-headed arrows represent directional effects and paths with double-headed arrows represent nondirectional covariances. Path coefficients are standardized, + p<.10 * p <.05; ** p < .01; *** p < .001.
Overall model fit: 2(3) = 1.03, p = .794, 2/df = .34, SRMR = .03, CFI = 1.00.
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